struction can afterwards be made as indicated above. If in equation (1), a2 + b2 = r2, the circle will pass through the vertex A and the equation, after dividing both members by y, will become y3 — (4pa — 4p3)y — 4bp2= 0, and the corresponding construction will give the roots of a cubic equation. In the equation of the fourth degree, if two of the roots are imaginary, that fact will be indicated by the circle only cutting the parabola in two points. Let it be required to construct the roots of the equation y2y4y2=0, which is of the required form. Equating the coefficients of the like powers of y in this and in equation (1), we have 24p2 - 4pa = 2p(2p - 2a), and − 2 = 2p(a2 + b2 — r2). Let us assume = 1: we deduce and r√21. - 4bp2 a = } b Which data enable us to make the con struction. M to each other, and set off on the line AB the Draw two lines AB and CD at right angles distances -OE and +OB, respectively equal to the inferior and superior limits of the real roots of the given equation. If the inferior limit is negative, as we have supposed in the figure, the distance OE will be laid off to the left; if it is positive, it must be laid off to the right. Assume, in succession, a sufficient number of values for z between the limits already determined, and substitute these separately for x in equation (2), and deduce the corresponding values of y. Each assumed value of x with the corresponding deduced value of y, will be the co-ordinates of a point which may be constructed by laying off the assumed value of x from O on the line AB, to the right, when positive, and to the left when negative. From the extremity of the distance laid off erect a perpendicular to AB, There are other methods of constructing and lay off on this perpendicular, from AB, roots of equations of the third and fourth the deduced value of y, observing that it must degrees, such as using an auxiliary ellipse, be laid off upwards if the value of y is posiconchoid, or cissoid. tive, and downwards if it is negative. 4. The construction of the roots of equations this manner a succession of points may be of a higher degree than the fourth is an determined, and a curve MPLQ traced operation which can only be approximately through them. performed. We shall simply indicate the The distances from O to the points in which general method of proceeding without making this curve cuts the line AB, will be the real any application of the principles developed. roots of the equation. The reason is appaLet us take an equation of the form rent, for when the curve whose equation is x+Bæm−1+Сxm-2+&c.+Nx+R=0..(1), equation (2) cuts the line AB, y must be equal in which B, C, &c, N, R, are known numto 0, and equation (2) for that value becomes bers. Find by the known rules of algebra equation (1), and these distances therefore the superior and inferior limits of the real represent the real roots. When the curve roots of the equation. Now let a second approaches the line AB, and then recedes equation be formed by placing y equal to the first member of the given equation, and from the principles of analytical geometry, the resulting equation y=xm+Bxm-1+Cxm−2+ &c. +Nx+R. . (2). will be the equation of a curve, which may be constructed approximately by points as follows: In from it without cutting it, as at L, such change of direction indicates a pair of imaginary roots. To insure as much accuracy as possible, great care should be taken in constructing the curve in the neighborhood of the points in which it cuts the line AB. Analogous methods may be employed for finding the values of the unknown quantities, when there are two equations containing two unknown quantities, the equations being nu- the particular case under the general one merical and of any degree whatever. Assume a pair of rectangular axes as before, and in like manner construct two curves by points. The first equation will be the equation of one curve, and the second equation will be that of the other curve. expressed in the formula, and then deducing the results. This will be the general formula to a particular binomial. The development of formulas and the deduction of rules, constitute the Science of Mathematics: the application of these to particular cases constitutes the Art of Mathematics. Most of the arts are little else than the application, either directly or indirectly, of the principles of science. Having constructed the curves, draw through the points of intersection straight lines parallel to the assumed axes. Then for each point there will be a pair of distances which will represent the simultaneous values of the unknown quantities. If the quantities are represented by y and x, as we have supposed, the distance to the axis AB will represent the value of y, and the distance to the axis CD will represent the simultaneous value of x. The number of points will determine the number of real solutions of the equations, and the accuracy of the values determined will depend upon the accuracy of the con-root cannot be obtained; but the longer the struction of the curves. If the given equations are both of the second degree, the curves to be constructed will be conic sections, and their construction may be more readily effected by some of the methods for constructing these lines. 3. APPLICATION OF GEOMETRY AND ALGEBRA TO TRIGONOMETRY. The subject of trigonometry is nothing more than a development of the results of applying the principles of geometry and algebra to determine the relation between angles and their functions. This subject is more fully discussed under the head of Trigonometry. The various applications of Mathematics to the physical sciences, engineering, &c., do not come within the scope of this work. The term application, in this sense, is used to denote the use which is made of the principles of mathematics in improving and developing these sciences. AP-PROX'I-MATE. [L. ad, to, and proximus, next]. In mathematics, an approximate result is one which is very near the true result; thus, the approximate value of a radical quantity is the result obtained by applying the rule for extracting the indicated root of the quantity under the radical sign, and continuing the operation to any desired extent. From the nature of the case, the true rule is applied, the more nearly will the result approximate to the true root. In short, the error may be reduced to less than any assignable quantity. The process of approximation is one of frequent use in all practical operations. See Approximation. AP-PROX-I-MA'TION. In mathematics, a method of calculation, by which we obtain an approximate value of a quantity which cannot be found accurately, either on account of the nature of the quantity itself, or on account of the imperfection of our mode of operation. The method of finding the ratio of the diameter of a circle to its circumference, or the length of the circumference of a circle whose diameter is 1, affords an instance of geometrical approximation. It is a principle of Geometry, that the arc of a circle is greater than its chord, however small the arc may be. Now, if we suppose a regular polygon, say of 64 sides, to be inscribed in a circle whose diameter is 1, it is evident that the length of the perimeter of the polygon will be an approximate value of the length of the circumference, though it will differ sensibly from it. If now we suppose that a regular polygon, of twice as many sides, is to be inscribed, the perimeter of the new polygon will approxiThe application of the binomial formula mate still more closely to the length of the in any case, consists in attributing to the let-circumference. If we continue to double the ters in the formula such values as will bring number of sides of the regular inscribed po APPLICATION OF A RULE OR FORMULA, consists in performing the operations prescribed by the rule, or indicated by the formula. Thus, the application of the rule for solving equations of the second degree, in any given case, consists in solving the particular equation by following the different steps prescribed in the rule, so as to determine the roots of the equation. value. lygon, we shall continue to approximate to circles, there may be an infinite number of the length of the circumference; but whatever sets of values of a, b, and r; but for a given may be the number of sides of the polygon, circle, and a given system of axes, a, b, and its perimeter will never be exactly equal tor become known, and are absolutely fixed in the circumference of the circle. The difference between the length of the perimeter and Not so, however, with the variables z and the circumference may be made less than any y. Whatever circle we choose to consider, assignable line, but it can never be made they will represent the co-ordinates of any equal to 0. point of its circumference at the same instant, and of every point in succession; that is, for any one set of values of the arbitrary constants, there is an infinite number of sets of values for the variables, which will satisfy the equation. In analysis, the attempt to express radical quantities in entire terms, affords an example of approximation, as also some of the methods of solving numerical equations of a higher degree than the fourth. In Arithmetic, the operation of converting certain vulgar fractions into equivalent decimal expressions, is one of approximation; thus 0.333333 · · 3. ad infinitum. What has been shown in this case, is in general true for all other cases; hence, the distinction between arbitrary constants and variables is this: given values may be attribuHere, no matter how far the division be car- ted, at pleasure, to the arbitrary constants, proried, the result will not express the exact vided they will satisfy the conditions of the value of, but for each decimal place added, the result will be a nearer approximation to its true value. In practical applications, the operation of approximation is one of great importance, as it gives results sufficiently accurate for the ordinary purposes of art. The various methods of finding approximate results will be fully described under the appropriate headings. problem, giving a particular case for each set of values. The variables, on the contrary, admit of every possible value which will satisfy the equation, in each and all the particular cases, determined by attributing given values to the constants. The use of arbitrary constants is to cause the equation under consideration to fulfill cer For tain conditions. The number of conditions which may be imposed, is, in general, equal ÄR'BI-TRA-RY. [L. arbitrarius, uncerto the number of arbitrary constants. tain, independent]. An arbitrary quantity in analysis is one to which we may assign any example: in the case already considered, we reasonable value at pleasure. In Analytical may cause the circle to pass through any Geometry, the arbitrary quantities are gener- the values of a, b, and r, so that the circle three points. The method of determining ally styled arbitrary constants, to distinguish shall pass through three given points, is to them from the variables which are in a certain sense arbitrary. Thus, in the general equation of the circle, the co-ordinates of each substitute, separately, for r and y, in the equa tion of the circle point. We thus obtain three equations of condition, which contain a, b, r and ≈ and y are variables, and a, b and r are known quantities; by combining these, we arbitrary constants. can find values of a, b and r, which, being substituted in the given equation, will make it the equation of a circle passing through the three given points. In the equation, a and b denote the co-ordinates of the centre, and by attributing to them suitable values, we may place the centre at any point of the co-ordinate plane. Since In the Integral Calculus, the constant, r denotes the radius, such a value may added to every integral obtained by applying be assigned to it as to give the circle any the rules for integration, is arbitrary in its desired extent. In this case, therefore, the nature, and serves to cause the integral to constants serve to determine the position and fulfill any reasonable condition. The method extent of the circle, with respect to the co- of using it for this end, is, to make such supordinate axes. positions upon the integral, as will cause it Since there may be an infinite number of to fulfill the required condition. We thus ob tain an equation from which we may deduce the value of the constant, which being substituted in the integral, will make it fulfill the required condition. For example, Sydx = X + C expresses a plane area included between the. curve of the axis of X and any two ordinates whatever. If now we wish that any given ordinate should limit the curve in one direction, we have simply to substitute 0 for the integral, since that integral is to commence at a given ordinate, and also to substitute for x, in X, a value a, corresponding to the given ordiThis gives nate. When there is but one intervening currency, it is called simple arbitration; when there is more than one it is called compound arbitration. The following is the rule for compound arbitration, and will answer also for simple arbitration: Multiply the sum to be converted by the following quotients, after canceling common factors, viz: A certain amount at the second place divided by its equivalent at the first; a certain amount at the third place divided by its equivalent at the second; a certain amount at the fourth place divided by its equivalent at the third place, and so on to the last place. In the above rule, the amounts named are supposed to be expressed in the currency of the place from which the remittance is made. If they are expressed in the currency of the place to which the remittance is made, the terms of the multipliers must be inverted. Example. A merchant in New York wishes to remit $4888,40 to London through Paris. ARC. [L. arcus, a bow]. A part of the circumference of a circle or other curve. When the term arc is used without any explanation, an arc of a circle is in general understood. As we have already explained, under angle, arcs of circles are employed as the measures of angles, in which case the centre of the arc is taken at the vertex of the angle. Where the radius is 1, the arc intercepted between the sides of the angle is taken as the measure of the angle; when the radius is not 1, the ratio of the radius to the intercepted arc is taken. There are various methods of expressing the values of angles by the aid of arcs of circles. Sometimes a portion of a circle, generally a quadrant, is assumed as a unit, and all other arcs are expressed numerically in terms of this as a standard; sometimes the whole circumference is divided into 360 equal parts, each of which is divided into 60 equal parts, which in turn are subdivided into 60 equal parts. These parts are called, respectively, degrees, minutes, and seconds, and the arcs are expressed in terms of these parts. There is no difference between these methods, except in the magnitude of the unit, and the manner of subdividing it. In expressing the magnitude of arcs, the radius is often taken as the unit, and since the circumference in that case is equal to 2, we may find the expression for any portion of a circumference, already expressed in degrees and fractions of a degree, by the following proportion : It is often convenient to express the length be moved so as constantly to touch the two of an arc in terms of its sine or tangent; outer nails, the vertex of the angle will trace this can only be done by means of series. out the arc of a circle between them. The most useful ones are subjoined: ARC'TIC. [Gr. арктоç, a bear]. The Arca3 3 as 3.5 a tic Circle is a circle of the sphere, whose sin-1aa+ + + + &c. plane passes through the north pole of the 2.3 2.4.5 2.4.6.7 ecliptic. It is about 661° distant from the tan-1a = a as a1 a9 all + 3 5 7 9 11 + &c. equator. ARE. [L. arca, an open surface]. In the decimal system of French measures the are is a square, the side of which is 10 metres in length. In contains 100 square metres or about 119.60 square yards. A'RE-X. [L. area, an open surface]. In geometry is the superficial contents of any surface expressed in terms of some given surface assumed as a unit or standard of comparison. The unit of measure is generally a square, one of whose sides is a linear unit in length. For different purposes, the area may be expressed in different terms. In land surveying, the areas of fields may be expressed in acres, the area of states may be given in square miles, whilst masons' and carpenters' work is generally expressed in square yards or square feet. In all cases, the arithmetical expression of an area is the ratio of some assumed surface to the surface in question. When surfaces are similar, they are to each other as the squares of their homologous lines. The most general formula for an area bounded by a plane curve, by the axis of X, and by any two ordinates, is 1. S= fydx; for an area of a surface generated by revolv 2. S = ƒ 2у√dx2 + dy2; 3. S d22 'dx dy 1 + To apply the first formula: + d22 dx2 dy3 AR-CHI-ME'DES' SPIRAL. See Spiral. ARC'O-GRAPH. [L. arcus, a bow, and Gr. ypapw, to describe]. An instrument used to describe an arc of a circle, without having its centre given. The simplest form is that used by carpenters for striking arcs Find from the equation of the curve, the for the top of doors, windows, &c. Three value of y in terms of x, and substitute it in nails being driven to mark three points of the the formula; then perform the integration indicircle, two pieces of board are nailed together, cated between any two limits, and the result forming an angle, so that their vertex shall be will give the area contained between the curve, at the middle nail, and the two sides against the axis of X, and the two ordinates taken as the extreme ones. If now the two pieces limits. |